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This video. Were going to talk about how to calculate the area of a a shader region. So lets say if we have a square inside of a and lets say.
The side length of the square is five inches and the rectangle lets say its 8. By 10 calculate the area of the shaded region to calculate the area of the shaded region you need to find the difference between the area of the large object and the area of the small object so in this example. The area of the shaded region is going to be the area of the rectangle minus.
The area of the square. The area of the rectangle is the left times the width. The area of the square is side squared.
Its in this case. The length of the rectangle is 10 the width is 8 and s. Is 5 for the square.
So its 10 times. 8. Minus.
5. Squared. So the area of the rectangle.
Is 80. The area of the square is 25 and 80 minus. 25.
Is 55. So the area of the shaded region. Is 55 square units or inches squared.
Now lets try another example so lets say if we have a square and theres a circle inside of the square and lets say the radius of the circle is 8 units. And lets say. The square is 20 units each side with this information.
Calculate the area of the shaded region feel free to pause. The video. If you want to so we have the side length of the square and the radius.
So the area of the shaded region is the area of the larger object. Which is the square minus. The area of the smaller object.
Which is the circle. Now we know the area of the square is s squared. The area of the circle is pi r.
Squared. So this is going to be xx squared minus. Pi times.
8. Squared. Now xx squared or 20 times.
20. Thats 400 8 times. 8.
Is 64. So the exact answer is 400. Minus 64.
Pi now lets get this answer in terms of a decimal and so this comes out to be 1. 989 square units. So thats the answer.
Lets try another simple problem before you start working on the harder problems. So. Lets say theres a circle within another circle and the inner radius.
Lets say its 4 units. Long. And lets say.
The outer radius is 7 units. Long. So with this information.
Calculate the area of the shaded region. So the area of the shaded region is the difference between the area of the large circle minus. The area of the small circle.
So for the large circle is going to be pi r squared. But using the outer radius and for the small circle. Its pi r squared.
But using the inner radius. So the outer radius is 7. The inner radius is 4 7.
Squared is 49 4. Squared is 16. So its 49 pi minus.
16. Pi. And thats going to be 33 pi.
So thats the area of the shaded region.
Thats the exact answer now lets try some harder problems. So lets say. This is the center of the circle.
And lets call this a b and c. So b. Is the center and its also a right angle and lets say the radius of the circle is 8.
What is the area of the shaded region. The area of the shaded region is going to be the difference between the area of the large object. Which is the circle minus.
The area of the small object. Which is the triangle. The area of a circle is pi r.
Squared and the area of a triangle is 1 2 base times height. So we have the radius notice that a b is the radius and also bc is the radius as well so a b and bc is a the distance between the center and any point on a circle is the radius of the circle. Now notice that the radius is also the base of the triangle and its also the height of the triangle as well so.
8. Squared is 64 and half of 64 is 32. So this is the area is 64 pi minus 32.
And so as a decimal thats 169 point one square units. So thats the answer heres another example so lets say if we have a rectangle and inside a rectangle. We have a rhombus now lets say this side of the rectangle is 8.
And this part of the rhombus is 5. What is the area of the shaded region go ahead and try this problem. So the area of the shaded region is the difference between the area of the rectangle and the area of the rhombus.
So lets draw the diagonals of the rectangle. The area of a rectangle is length times. Width.
The area of a rhombus is 1 2 or d1 times d2. So d1 is basically the same as the width of the rectangle and d2. Is the same as the length of the rectangle in this example.
So im going to replace d1 with w. And deets you with allen. So lw minus 1.
2. Lw. Will just be 1 2.
Lw. So we have the width already w. Is 8.
What we need to do is calculate the lef and then we can calculate the area of the shaded region. So how can we do this well we need to know that the diagonals of a rhombus. They meet at right angles.
And also they bisect each other so these two sides are congruent and those two segments are congruent. So therefore. Its d1 is 8.
That means that these two sides. Therefore. Lets call.
This. A b c. D.
E. So now we need to calculate segment ae. And ec or just.
Easy. So notice that we have a right triangle. So we could find a mess inside this side is 4.
This side is 5. So lets calculate a c. Squared is equal to a squared plus b.
Squared so c. Is. 5 and b.
Is. 4. So.
5. Squared is 25 4. Squared is 16 25 minus.
16. Is 9. And so if we take the square root of both sides aah.
3. So now we have everything that we need in order to calculate the area so. If ec history.
Is also three which means that the left of the triangle. I mean the left of the rectangle is 6 3. Plus.
3 is 6. So the area is going to be 1 2. Times.
6 times 8 half of 6 is 3 3. Times. 8.
Is 24. So the area is 24 square units. Now lets say if we have a circle and lets say theres a triangle around a circle.
Now lets say the radius of the circle is 20. What is the area of the shaded region now last time when we had a circle and a rectangle. The rectangle was the square because the circle is even all around so this type of triangle around a circle has to be a equilateral triangle and notice that if you draw a line where it touches the triangle and a circle thats r.
And notice is the same everywhere for all three points. Where the circle meets. The triangle.
So you need to realize that this is an equilateral triangle and so if we can calculate s. We can calculate the area of the equilateral triangle. So r is 20.
Lets turn this into a right triangle. So this side is 20 how can we calculate this part. Because if this is s.
Then this part is 1 2. Of us all we need is an angle. If we could find one of these two angles.
Then we can do it so. The question is how can we find one of those angles. So notice that if we draw a line between the center and the vertex of the triangle.
We can draw three of such lines and the angle of a full circle is 360. So if you take 360 and divided by 3 that will give you 120. Which is this angle.
And thats also this angle. So therefore this angle. Here must be half of 120 which is 60 so if this angle is 60.
The other one is 30. So we have a 30 60 90 right triangle now across the 30. Lets say.
This is to the side across the 30 is going to be half of the hypotenuse. So thats going to be 1 and aside across the 60 is whatever this value is times the square root of 3. Now we have this side across the 20.
I mean across the 30 which is 20. So the hypotenuse is twice that value its 40 and so side across the 60 is whatever this is times the square root of three so its 20 square root. 3.
And so thats thats over 2. So if thats half of s s. Itself has to be twice the value so s is 40 square root.
3. And so now we have enough information to calculate the area of the shaded region. The area of the shaded region is going to be the area of the triangle minus.
The area of the circle. The area of an equilateral triangle is the square root of 3 4. Times s.
Squared and the area of a circle. Is pi r. Squared so s is 40 square root.
3. And r rs. 20.
That was given to us at the beginning now 40 squared thats 40 times. 40. Thats 1600 and the square root of.
3. Squared is just 3 20. Times.
20 is. 400 now 1600. Divided by 4.
Thats 400 and 400 times. 3. Is 12 hundred.
So the area is going to be 1200 square root. 3. Minus 400 pod.
So this is the exact answer for the area of the shaded region. .
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